Ann. Mat. Pura Appl. 186 (2007), 317--339

Christian Kassel and Christophe Reutenauer

Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2

Mathematics Subject Classification (2000): 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15

Abstract. We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group of automorphisms of the rank two free group F_2 and show that it can be realized as a monoid in the group of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F_2 lifting any given basis of the free abelian group Z+Z. We further give an algorithm allowing to decide whether two elements of F_2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.

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(15 mars 2007)